Intersection homology II (original) (raw)
Refined intersection homology on non-Witt spaces
Journal of Topology and Analysis, 2015
We investigate a generalization to non-Witt stratified spaces of the intersection homology theory of Goresky–MacPherson. The second-named author has described the self-dual sheaves compatible with intersection homology, and the other authors have described a generalization of Cheeger's L2 de Rham cohomology. In this paper we first extend both of these cohomology theories by describing all sheaf complexes in the derived category of constructible sheaves that are compatible with middle perversity intersection cohomology, though not necessarily self-dual. Our main result is that this refined intersection cohomology theory coincides with the analytic de Rham theory on Thom–Mather stratified spaces. The word "refined" is motivated by the fact that the definition of this cohomology theory depends on the choice of an additional structure (mezzo-perversity) which is automatically zero in the case of a Witt space.
Homology stratifications and intersection homology
Proceedings of the Kirbyfest, 1999
A homology stratification is a filtered space with local homology groups constant on strata. Despite being used by Goresky and MacPherson [3] in their proof of topological invariance of intersection homology, homology stratifications do not appear to have been studied in any detail and their properties remain obscure. Here we use them to present a simplified version of the Goresky-MacPherson proof valid for PL spaces, and we ask a number of questions. The proof uses a new technique, homology general position, which sheds light on the (open) problem of defining generalised intersection homology.
Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry, 2022
for all p ≥ 0. Furthermore, this assignment is functorial (more precisely, it is a functor from the category of cochain complexes and chain maps to the category of abelian groups and their homomorphisms). At first glance cohomology appears to be very abstract so it is natural to look for explicit examples. A way to obtain a cochain complex is to apply the operator (functor) Hom Z (-, G) to a chain complex C, where G is any abelian group. Given a fixed abelian group A, for any abelian group B we denote by Hom Z (B, A) the abelian group of all homomorphisms from B to A. Given any two abelian groups B and C, for any homomorphism f : The map Hom Z (f, A) is also denoted by Hom Z (f, id A ) or even Hom Z (f, id). Observe that the effect of Hom Z (f, id) on ϕ is to precompose ϕ with f . If f : B → C and g : C → D are homomorphisms of abelian groups, a simple computation shows that Hom R (g 1.1. EXACT SEQUENCES, CHAIN COMPLEXES, HOMOLOGY, COHOMOLOGY 21 Observe that Hom Z (f, id) and Hom Z (g, id) are composed in the reverse order of the composition of f and g. It is also immediately verified that Hom Z (id A , id) = id Hom Z (A,G) . We say that Hom Z (-, id) is a contravariant functor (from the category of abelian groups and group homomorphisms to itself). Then given a chain complex we can form the cochain complex obtained by applying Hom Z (-, G), and denoted Hom Z (C, G). The coboundary map d p is given by which means that for any f ∈ Hom Z (C p , G), we have Thus, for any (p + 1)-chain c ∈ C p+1 we have We obtain the cohomology groups H p (Hom Z (C, G)) associated with the cochain complex Hom Z (C, G). The cohomology groups H p (Hom Z (C, G)) are also denoted H p (C; G). This process was applied to the simplicial chain complex C * (K) associated with a simplicial complex K by Alexander and Kolmogoroff to obtain the simplicial cochain complex Hom Z (C * (K); G) denoted C * (K; G) and the simplicial cohomology groups H p (K; G) of the simplicial complex K; see Section 5.6. Soon after, this process was applied to the singular chain complex S * (X; Z) of a space X to obtain the singular cochain complex Hom Z (S * (X; Z); G) denoted S * (X; G) and the singular cohomology groups H p (X; G) of the space X; see Section 4.8. Given a continuous map f : X → Y , there is an induced chain map f : S * (Y ; G) → S * (X; G) between the singular cochain complexes S * (Y ; G) and S * (X; G), and thus homomorphisms of cohomology f * : H p (Y ; G) → H p (X; G). Observe the reversal: f is a map from X to Y , but f * maps H p (Y ; G) to H p (X; G). We say that the map X → (H p (X; G)) p≥0 is a contravariant functor from the category of topological spaces and continuous maps to the category of abelian groups and their homomorphisms. So far our homology groups have coefficients in Z, but the process of forming a cochain complex Hom Z (C, G) from a chain complex C allows the use of coefficients in any abelian obtained by applying Hom R (-, G), and denoted Hom R (C, G).
Intersection Cohomology on Nonrational Polytopes⋆
Compositio Mathematica, 2003
Abstract. We consider a fan as a ringed space (with finitely many points). We develop the cor-responding sheaf theory and functors, such as direct image Rp√ (p is a subdivision of a fan), Verdier duality, etc. The distinguished sheaf LF, called the minimal sheaf plays the ...
The hypertoric intersection cohomology ring
Inventiones Mathematicae, 2009
We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset.
Intersection cohomology of pure sheaf spaces using Kirwan’s desingularization
Journal of Geometry and Physics
Let Mn be the Simpson compactification of twisted ideal sheaves IL,Q(1) where Q is a rank 4 quardric hypersurface in P n and L is a linear subspace of dimension n − 2. This paper calculates the intersection Poincaré polynomial of Mn using Kirwan's desingularization method. We obtain the intersection Poincaré polynomial of the moduli space for one-dimensional sheaves on del Pezzo surfaces of degree ≥ 8 by considering wall-crossings of stable pairs and complexes.
2006
The contents of these notes originated from a talk delivered to a diversified audience at the Departamento de Matemática of my own University. The present notes expand and give full details of statements made in that occasion. Moreover, I have sought to expand as well on several algebraic notions in order to pave the way towards a more stable version even though self-sufficiency is out of question in a material of this order. If one is to single a module that stands alone in its totally device-independent nature, one'd better take the module of differentials Ω(A/k). In fact, most invariants of an algebra or scheme are attached primevally to this module or "découlent" from it by various procedures. Thus, Segre classes in algebraic geometry are ultimately defined in terms of Ω(A/k) and even the ubiquitous canonical module (dualizing sheaf) is "normally" related to Ω(A/k) by double-dualizing its top-wedge module. My objective in this Course is panorama-driven, due to the short span of lectures. However, details and examples will be worked out and, possibly, a few arguments too. 10. * Considerations about a formula of Kleiman-Plücker-Teissier for the degree (class) of the dual variety to a complete intersection.